problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 6
values | synthetic bool 1
class |
|---|---|---|---|---|---|---|---|---|
Find the sum of the numbers written with two digits $\overline{ab}$ for which the equation $3^{x + y} =3^x + 3^y + \overline{ab}$ has at least one solution $(x, y)$ in natural numbers. | 1. We start with the equation \(3^{x + y} = 3^x + 3^y + \overline{ab}\), where \(\overline{ab}\) is a two-digit number, and \(x, y\) are natural numbers.
2. First, we note that if both \(x\) and \(y\) are 0 or 1, the equation cannot hold because the left-hand side (LHS) would be too small compared to the right-hand si... | 78 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$\frac{a^3}{b^3}$+$\frac{a^3+1}{b^3+1}$+...+$\frac{a^3+2015}{b^3+2015}$=2016
b - positive integer, b can't be 0
a - real
Find $\frac{a^3}{b^3}$*$\frac{a^3+1}{b^3+1}$*...*$\frac{a^3+2015}{b^3+2015}$
| 1. Given the equation:
\[
\frac{a^3}{b^3} + \frac{a^3+1}{b^3+1} + \cdots + \frac{a^3+2015}{b^3+2015} = 2016
\]
where \( b \) is a positive integer and \( a \) is a real number.
2. We need to analyze the behavior of the terms \(\frac{a^3+n}{b^3+n}\) for \( n = 0, 1, 2, \ldots, 2015 \).
3. Consider the case... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Real numbers $a$ and $b$ satisfy the system of equations $$\begin{cases} a^3-a^2+a-5=0 \\ b^3-2b^2+2b+4=0 \end{cases}$$ Find the numerical value of the sum $a+ b$. | Given the system of equations:
\[
\begin{cases}
a^3 - a^2 + a - 5 = 0 \\
b^3 - 2b^2 + 2b + 4 = 0
\end{cases}
\]
we need to find the numerical value of the sum \(a + b\).
1. Let \(a + b = k\). Then \(b = k - a\).
2. Substitute \(b = k - a\) into the second equation:
\[
(k - a)^3 - 2(k - a)^2 + 2(k - a) + 4 = 0
\]
3. ... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sets $A_1=\{1\}$, $A_2=\{2,3,4\}$, $A_3=\{5,6,7,8,9\}$, etc. Let $b_n$ be the arithmetic mean of the smallest and the greatest element in $A_n$. Show that the number $\frac{2000}{b_1-1}+\frac{2000}{b_2-1}+\ldots+\frac{2000}{b_{2000}-1}$ is a prime integer. | 1. **Identify the structure of the sets \( A_n \):**
- The set \( A_1 = \{1\} \) has 1 element.
- The set \( A_2 = \{2, 3, 4\} \) has 3 elements.
- The set \( A_3 = \{5, 6, 7, 8, 9\} \) has 5 elements.
- Generally, the set \( A_n \) has \( 2n-1 \) elements.
2. **Determine the smallest and largest elements ... | 1999 | Number Theory | proof | Yes | Yes | aops_forum | false |
For any $n\in\mathbb N$, denote by $a_n$ the sum $2+22+222+\cdots+22\ldots2$, where the last summand consists of $n$ digits of $2$. Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$. | 1. **Define the sequence \(a_n\)**:
For any \(n \in \mathbb{N}\), \(a_n\) is defined as the sum \(2 + 22 + 222 + \cdots + \underbrace{22\ldots2}_{n \text{ digits}}\).
2. **Express \(a_n\) in a general form**:
Each term in the sum can be written as \(2 \times \frac{10^k - 1}{9}\) for \(k = 1, 2, \ldots, n\). Ther... | 222 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all permutations of the numbers $1,2,\ldots,9$ in which no two adjacent numbers have a sum divisible by $7$ or $13$.
| To solve the problem of finding all permutations of the numbers \(1, 2, \ldots, 9\) such that no two adjacent numbers have a sum divisible by \(7\) or \(13\), we need to carefully analyze the constraints and use combinatorial methods.
1. **Identify the pairs that sum to \(7\) and \(13\):**
- For \(7\):
\[
... | 74880 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A box $3\times5\times7$ is divided into unit cube cells. In each of the cells, there is a c[i][/i]ockchafer. At a signal, every c[i][/i]ockchafer moves through a face of its cell to a neighboring cell.
(a) What is the minimum number of empty cells after the signal?
(b) The same question, assuming that the c[i][/i]ockc... | ### Part (a)
1. **Coloring the Cells:**
- Consider a $3 \times 5 \times 7$ box divided into unit cubes.
- Color the cells in a 3D checkerboard pattern, with black in the corners. This means that cells at coordinates $(x, y, z)$ where $x + y + z$ is even are black, and those where $x + y + z$ is odd are white.
2.... | 35 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a positive integer $ k$, there is a positive integer $ n$ with the property that one can obtain the sum of the first $ n$ positive integers by writing some $ k$ digits to the right of $ n$. Find the remainder of $ n$ when dividing at $ 9$. | 1. We start by noting that the sum of the first $n$ positive integers is given by the formula:
\[
S = \frac{n(n+1)}{2}
\]
2. According to the problem, there exists a positive integer $n$ such that appending $k$ digits to the right of $n$ results in the sum of the first $n$ positive integers. This can be expres... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by
$F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$. Find
the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \] | 1. **Define the Fibonacci sequence and the problem:**
The Fibonacci sequence \((F_n)_{n \in \mathbb{N}^*}\) is defined by:
\[
F_1 = 1, \quad F_2 = 1, \quad F_{n+1} = F_n + F_{n-1} \quad \text{for every } n \geq 2.
\]
We need to find the limit:
\[
\lim_{n \to \infty} \left( \sum_{i=1}^n \frac{F_i}{2... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Define the sequence $(x_{n})$: $x_{1}=\frac{1}{3}$ and $x_{n+1}=x_{n}^{2}+x_{n}$. Find $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$, wehere $[$ $]$ denotes the integer part. | 1. We start with the given sequence \( (x_n) \) defined by \( x_1 = \frac{1}{3} \) and \( x_{n+1} = x_n^2 + x_n \). We can rewrite the recurrence relation as:
\[
x_{n+1} = x_n (x_n + 1)
\]
2. We need to find the sum \( \left\lfloor \sum_{k=1}^{2007} \frac{1}{x_k + 1} \right\rfloor \), where \( \lfloor \cdot \... | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$. | 1. **Substitute \( X = e^{2\pi i \phi} \) into the polynomial \( P(X) \):**
\[
P(X) = \sqrt{3} \cdot X^{n+1} - X^n - 1
\]
becomes
\[
P(e^{2\pi i \phi}) = \sqrt{3} \cdot e^{2\pi i \phi (n+1)} - e^{2\pi i \phi n} - 1
\]
2. **Express the exponents in terms of \(\phi\):**
\[
\sqrt{3} \cdot e^{2\... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ (a_{n})_{n\ge 1} $ be a sequence such that: $ a_{1}=1; a_{n+1}=\frac{n}{a_{n}+1}.$ Find $ [a_{2008}] $ | 1. **Initial Setup and Recurrence Relation:**
Given the sequence \( (a_n)_{n \ge 1} \) with \( a_1 = 1 \) and the recurrence relation:
\[
a_{n+1} = \frac{n}{a_n + 1}
\]
We need to find \( [a_{2008}] \), where \([x]\) denotes the floor function of \(x\).
2. **Rationality of the Sequence:**
Since \( a_... | 2007 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Solve in positive integers the following equation $$\left [\sqrt{1}\right]+\left [\sqrt{2}\right]+\left [\sqrt{3}\right]+\ldots+\left [\sqrt{x^2-2}\right]+\left [\sqrt{x^2-1}\right]=125,$$ where $[a]$ is the integer part of the real number $a$. | 1. **Understanding the Problem:**
We need to solve the equation
\[
\left \lfloor \sqrt{1} \right \rfloor + \left \lfloor \sqrt{2} \right \rfloor + \left \lfloor \sqrt{3} \right \rfloor + \ldots + \left \lfloor \sqrt{x^2-2} \right \rfloor + \left \lfloor \sqrt{x^2-1} \right \rfloor = 125,
\]
where $\left... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations. | 1. **Claim and Example:**
We claim that the maximum value of \(\frac{AP}{PE}\) is \(\boxed{5}\). This is achieved by letting \(A = (0, 0)\), \(B = (3, 0)\), and \(C = (2, 2)\). In this case, \(P = (2, 1)\) and \(\frac{AP}{PE} = 5\).
2. **Optimality Proof:**
Let \(x, y, z\) be the areas of \(\triangle BPC\), \(\t... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Hello Everyone,
i'm trying to make a strong marathon for number theory .. which will be in Pre-Olympiad level
Please if you write any problem don't forget to indicate its number and if you write a solution please indicate for what problem also to prevent the confusion that happens in some marathons.
it will be pre... | To solve the problem, we need to find \( f(f(f(N))) \) where \( N = 4444^{4444} \) and \( f(n) \) denotes the sum of the digits of \( n \).
1. **Understanding the Sum of Digits Function \( f(n) \)**:
The function \( f(n) \) represents the sum of the digits of \( n \). A key property of this function is that \( f(n)... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
\[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\]
where $n$ is an positive integer. Find the largest $k$ such that $n^k$ divides $[A]$. | 1. We start with the given expression for \( A \):
\[
A = 3 \sum_{m=1}^{n^2} \left( \frac{1}{2} - \{\sqrt{m}\} \right)
\]
where \( \{x\} \) denotes the fractional part of \( x \).
2. We decompose the sum:
\[
A = 3 \left( \sum_{m=1}^{n^2} \frac{1}{2} - \sum_{m=1}^{n^2} \{\sqrt{m}\} \right)
\]
Th... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $m$ circles intersect in points $A$ and $B$. We write numbers using the following algorithm: we write $1$ in points $A$ and $B$, in every midpoint of the open arc $AB$ we write $2$, then between every two numbers written in the midpoint we write their sum and so on repeating $n$ times. Let $r(n,m)$
be the number o... | To solve the problem, we need to determine the number of appearances of the number \( n \) in the sequences generated by the algorithm described. We will also find the smallest \( m \) for which \( r(2006, m) \) is a perfect square.
### Part (a): Determine \( r(n, m) \)
1. **Understanding the Sequence Generation:**
... | 841 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A non-empty set $ S$ of positive integers is said to be [i]good[/i] if there is a coloring with $ 2008$ colors of all positive integers so that no number in $ S$ is the sum of two different positive integers (not necessarily in $ S$) of the same color. Find the largest value $ t$ can take so that the set $ S\equal{}\{a... | To solve this problem, we need to find the largest value \( t \) such that the set \( S = \{a+1, a+2, \ldots, a+t\} \) is *good*. A set \( S \) is *good* if there exists a coloring with 2008 colors of all positive integers such that no number in \( S \) is the sum of two different positive integers of the same color.
... | 4014 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The function $f:\mathbb{N} \rightarrow \mathbb{N}$ verifies:
$1) f(n+2)-2022 \cdot f(n+1)+2021 \cdot f(n)=0, \forall n \in \mathbb{N};$
$2) f(20^{22})=f(22^{20});$
$3) f(2021)=2022$.
Find all possible values of $f(2022)$. | 1. **Define the sequence and the recurrence relation:**
Let \( a_n \triangleq f(n) \). The given recurrence relation is:
\[
a_{n+2} - 2022 \cdot a_{n+1} + 2021 \cdot a_n = 0.
\]
2. **Find the characteristic equation:**
The characteristic equation associated with this recurrence relation is:
\[
\la... | 2022 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $f:\mathbb{N} \rightarrow \mathbb{N},$ $f(n)=n^2-69n+2250$ be a function. Find the prime number $p$, for which the sum of the digits of the number $f(p^2+32)$ is as small as possible. | 1. **Rewrite the function and analyze modulo 9:**
Given the function \( f(n) = n^2 - 69n + 2250 \), we want to find the prime number \( p \) such that the sum of the digits of \( f(p^2 + 32) \) is minimized.
First, we simplify \( f(n) \) modulo 9:
\[
f(n) \equiv n^2 - 69n + 2250 \pmod{9}
\]
Since \( ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3? | To solve the problem of filling a \(4 \times 4\) board with nonnegative integers such that the sum of the numbers in each row and each column is 3, we need to consider the possible distributions of the numbers in each row and column.
1. **Identify possible row/column configurations:**
- If four non-negative intege... | 2008 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The plane is divided into unit cells, and each of the cells is painted in one of two given colors. Find the minimum possible number of cells in a figure consisting of entire cells which contains each of the $16$ possible colored $2\times2$ squares. | 1. **Identify the problem**: We need to find the minimum number of cells in a figure that contains all possible $2 \times 2$ colored squares, given that each cell is painted in one of two colors.
2. **Understand the $2 \times 2$ squares**: There are $2^4 = 16$ possible $2 \times 2$ squares since each of the 4 cells in... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a country, there are $2018$ cities, some of which are connected by roads. Each city is connected to at least three other cities. It is possible to travel from any city to any other city using one or more roads. For each pair of cities, consider the shortest route between these two cities. What is the greatest number... | 1. **Understanding the Problem:**
- We have 2018 cities, each connected to at least three other cities.
- The graph is connected, meaning there is a path between any two cities.
- We need to find the greatest number of roads (edges) that can be on the shortest route between any two cities.
2. **Clarifying the... | 1511 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For an integer $n \ge 3$ we consider a circle with $n$ points on it.
We place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called [i]stable [/i] as three numbers next to always have product $n$ each other.
For how many values of $n$ with $3 \l... | 1. **Understanding the Problem:**
We need to determine for how many values of \( n \) with \( 3 \le n \le 2020 \) it is possible to place numbers in a stable way on a circle with \( n \) points. A placement is stable if the product of any three consecutive numbers is equal to \( n \).
2. **Analyzing the Stability C... | 680 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $300$ participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of $n$ for which ... | 1. **Interpret the problem as a graph theory problem:**
- Consider each participant as a vertex in a graph.
- Each game of chess between two participants is represented by an edge between two vertices.
- The condition that no three contestants play against each other translates to the graph being $K_3$-free (n... | 200 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For real numbers $x$ and $y$ we define $M(x, y)$ to be the maximum of the three numbers $xy$, $(x- 1)(y - 1)$, and $x + y - 2xy$. Determine the smallest possible value of $M(x, y)$ where $x$ and $y$ range over all real numbers satisfying $0 \le x, y \le 1$. | To determine the smallest possible value of \( M(x, y) \) where \( 0 \le x, y \le 1 \), we need to analyze the three expressions \( xy \), \( (x-1)(y-1) \), and \( x + y - 2xy \).
1. **Define the expressions:**
\[
f(x, y) = xy
\]
\[
g(x, y) = (x-1)(y-1)
\]
\[
h(x, y) = x + y - 2xy
\]
2. **A... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In a sequence $a_1, a_2, . . . , a_{1000}$ consisting of $1000$ distinct numbers a pair $(a_i, a_j )$ with $i < j$ is called [i]ascending [/i] if $a_i < a_j$ and [i]descending[/i] if $a_i > a_j$ . Determine the largest positive integer $k$ with the property that every sequence of $1000$ distinct numbers has at least $k... | To solve this problem, we need to determine the largest positive integer \( k \) such that every sequence of 1000 distinct numbers has at least \( k \) non-overlapping ascending pairs or at least \( k \) non-overlapping descending pairs.
1. **Understanding the Problem:**
- We have a sequence of 1000 distinct number... | 333 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest $n$ such that $n \equiv (a - 1)$ mod $a$ for all $a \in \{2,3,..., 10\}$. | 1. We need to determine the smallest \( n \) such that \( n \equiv (a - 1) \pmod{a} \) for all \( a \in \{2, 3, \ldots, 10\} \).
2. This condition implies that \( n \equiv -1 \pmod{a} \) for each \( a \) in the set \(\{2, 3, \ldots, 10\}\).
3. To satisfy \( n \equiv -1 \pmod{a} \) for all \( a \), \( n + 1 \) must be a... | 2519 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Prove that if $ n$ is an odd positive integer, then the last two digits of $ 2^{2n}(2^{2n\plus{}1}\minus{}1)$ in base $ 10$ are $ 28$. | 1. Let \( n \) be an odd positive integer. We need to show that the last two digits of \( 2^{2n}(2^{2n+1} - 1) \) in base 10 are 28. This is equivalent to proving:
\[
2^{2n}(2^{2n+1} - 1) \equiv 28 \pmod{100}
\]
2. Since \( n \) is odd, we can write \( n = 2k + 1 \) for some integer \( k \). Substituting \( n... | 28 | Number Theory | proof | Yes | Yes | aops_forum | false |
Consider the sequence $ a_1\equal{}\frac{3}{2}, a_{n\plus{}1}\equal{}\frac{3a_n^2\plus{}4a_n\minus{}3}{4a_n^2}.$
$ (a)$ Prove that $ 1<a_n$ and $ a_{n\plus{}1}<a_n$ for all $ n$.
$ (b)$ From $ (a)$ it follows that $ \displaystyle\lim_{n\to\infty}a_n$ exists. Find this limit.
$ (c)$ Determine $ \displaystyle\lim_{n\t... | 1. **Prove that \(1 < a_n < 3\) for all positive integers \(n\)**:
- **Base Case**: For \(n = 1\), we have \(a_1 = \frac{3}{2}\). Clearly, \(1 < \frac{3}{2} < 3\).
- **Inductive Step**: Assume \(1 < a_n < 3\) for some \(n\). We need to show that \(1 < a_{n+1} < 3\).
- **Prove \(a_{n+1} > 1\)**:
\[
... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A $9 \times 9$ square consists of $81$ unit squares. Some of these unit squares are painted black, and the others are painted white, such that each $2 \times 3$ rectangle and each $3 \times 2$ rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares. | 1. **Understanding the Problem:**
We are given a $9 \times 9$ grid with some unit squares painted black and others painted white. Each $2 \times 3$ and $3 \times 2$ rectangle within this grid contains exactly 2 black unit squares and 4 white unit squares. We need to determine the total number of black unit squares i... | 27 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A right triangle with perpendicular sides $a$ and $b$ and hypotenuse $c$ has the following properties:
$a = p^m$ and $b = q^n$ with $p$ and $q$ prime numbers and $m$ and $n$ positive integers, $c = 2k +1$ with $k$ a positive integer.
Determine all possible values of $c$ and the associated values of $a$ and $b$. | 1. Given a right triangle with perpendicular sides \(a\) and \(b\) and hypotenuse \(c\), we have the following properties:
\[
a = p^m \quad \text{and} \quad b = q^n
\]
where \(p\) and \(q\) are prime numbers, \(m\) and \(n\) are positive integers, and \(c = 2k + 1\) with \(k\) a positive integer.
2. Accord... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can one choose distinct numbers a and b from {1, 2, 3, ..., 2005} such that a + b is a multiple of 5? | 1. **Identify the set and the condition:**
We need to choose distinct numbers \(a\) and \(b\) from the set \(\{1, 2, 3, \ldots, 2005\}\) such that \(a + b\) is a multiple of 5.
2. **Consider the numbers modulo 5:**
The numbers in the set \(\{1, 2, 3, \ldots, 2005\}\) can be classified based on their residues mod... | 401802 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \in \{1,2,3,4,5\}$ and $i \neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$? | 1. **Define the problem and the sums:**
We are given five distinct real numbers \(a_1, a_2, a_3, a_4, a_5\). We need to consider all possible sums of the form \(a_i + a_j\) where \(i \neq j\). There are \(\binom{5}{2} = 10\) such sums since each pair \((i, j)\) is considered once.
2. **Calculate the number of disti... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$1+2+3+4+5+6=6+7+8$.
What is the smallest number $k$ greater than $6$ for which:
$1 + 2 +...+ k = k + (k+1) +...+ n$, with $n$ an integer greater than $k$ ? | 1. We start with the given equation:
\[
1 + 2 + 3 + \cdots + k = k + (k+1) + (k+2) + \cdots + n
\]
We need to find the smallest number \( k \) greater than 6 for which this equation holds, with \( n \) being an integer greater than \( k \).
2. First, we use the formula for the sum of the first \( k \) natu... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of integers $a$ satisfying $1 \le a \le 100$ such that $a^a$ is a perfect square.
(And prove that your answer is correct.) | To determine the number of integers \( a \) satisfying \( 1 \le a \le 100 \) such that \( a^a \) is a perfect square, we need to analyze the conditions under which \( a^a \) can be a perfect square.
1. **Condition Analysis**:
- For \( a^a \) to be a perfect square, \( a \) itself must either be a perfect square or ... | 55 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer n is called [i]primary divisor [/i] if for every positive divisor $d$ of $n$ at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is divisor primary, because its positive divisors $1$, $2$, $4$, and $8$ each differ by $1$ from a prime number ($2$, $3$, $5$, and $7$, respectively... | To determine the largest primary divisor number, we need to verify the definition and properties of primary divisors. A positive integer \( n \) is called a primary divisor if for every positive divisor \( d \) of \( n \), at least one of the numbers \( d - 1 \) and \( d + 1 \) is prime.
1. **Lemma: Every odd number \... | 48 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Kira has $3$ blocks with the letter $A$, $3$ blocks with the letter $B$, and $3$ blocks with the letter $C$. She puts these $9$ blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence $ABCAABCBC$ the blocks with the letter A have d... | To solve this problem, we need to determine the maximum number of distinct distances between blocks with the same letter in a sequence of 9 blocks, where each letter (A, B, C) appears exactly 3 times.
1. **Understanding the Problem:**
- We have 9 blocks in total: 3 A's, 3 B's, and 3 C's.
- We need to find the ma... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In square $ABCD$, $\overline{AC}$ and $\overline{BD}$ meet at point $E$.
Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$.
If $\overline{AF}$ meets $\overline{ED}$ at point $G$, and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$. | 1. **Identify the properties of the square and the given points:**
- Let the side length of the square \(ABCD\) be \(s\).
- The diagonals \(\overline{AC}\) and \(\overline{BD}\) intersect at point \(E\), which is the center of the square.
- Since \(E\) is the center, \(E\) divides each diagonal into two equal ... | 48 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If $m, n$, and $p$ are three different natural numbers, each between $2$ and $9$, what then are all the possible integer value(s) of the expression $\frac{m+n+p}{m+n}$? | 1. We start with the given expression $\frac{m+n+p}{m+n}$, where $m, n$, and $p$ are distinct natural numbers between $2$ and $9$.
2. First, we note that $m$ and $n$ are distinct natural numbers between $2$ and $9$. Therefore, the minimum value of $m+n$ is $2+3=5$ (since $m$ and $n$ are distinct), and the maximum valu... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In two weeks three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks. In four weeks, two cows eat all the grass on two hectares of land, together with all the grass that regrows there during the four weeks.
How many cows will eat all the grass on six he... | 1. Let \( x \) be the amount of hectares that a cow eats per week, and \( y \) be the amount of hectares of grass that grows per week.
2. From the problem, we have two scenarios:
- In two weeks, three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive value of $36^k - 5^m$, where $k$ and $m$ are positive integers. | 1. **Understanding the problem**: We need to find the smallest positive value of \(36^k - 5^m\) where \(k\) and \(m\) are positive integers.
2. **Modulo 5 analysis**:
- Note that \(36 \equiv 1 \pmod{5}\). Therefore, \(36^k \equiv 1^k \equiv 1 \pmod{5}\).
- This implies \(36^k - 5^m \equiv 1 - 0 \equiv 1 \pmod{5... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A rectangular sheet of paper whose dimensions are $12 \times 18$ is folded along a diagonal, creating the $M$-shaped region drawn in the picture (see below). Find the area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/4/7/d82cde3e91ab83fa14cd6cefe9bba28462dde1.png[/img] | 1. **Understanding the Problem:**
We have a rectangular sheet of paper with dimensions \(12 \times 18\). When folded along a diagonal, it creates an \(M\)-shaped region. We need to find the area of the shaded region.
2. **Identify the Diagonal:**
The diagonal of the rectangle can be calculated using the Pythagor... | 138 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Starting with a list of three numbers, the “[i]Make-My-Day[/i]” procedure creates a new list by replacing each number by the sum of the other two. For example, from $\{1, 3, 8\}$ “[i]Make-My-Day[/i]” gives $\{11, 9, 4\}$ and a new “[i]MakeMy-Day[/i]” leads to $\{13, 15, 20\}$. If we begin with $\{20, 1, 8\}$, what is t... | 1. Let's denote the initial list of numbers as $(a, b, c)$. The "Make-My-Day" procedure transforms this list into $(b+c, a+c, a+b)$.
2. We need to find the maximum difference between any two numbers in the list after applying the "Make-My-Day" procedure multiple times.
3. First, let's analyze the transformation:
\[... | 19 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There is a sequence of numbers $+1$ and $-1$ of length $n$. It is known that the sum of every $10$ neighbouring numbers in the sequence is $0$ and that the sum of every $12$ neighbouring numbers in the sequence is not zero. What is the maximal value of $n$? | 1. **Claim**: The maximal value of \( n \) is 15. We will first verify that \( n = 15 \) is possible and then show that \( n \geq 16 \) is not possible.
2. **Verification for \( n = 15 \)**:
Consider the sequence:
\[
-1, -1, -1, -1, -1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1
\]
- The sum of any 10 consecutiv... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$, where $a$, $ b$, $c$, $d$ are non-negative integers.
| 1. We start by considering the expression $\frac{2^a - 2^b}{2^c - 2^d}$, where $a, b, c, d$ are non-negative integers. Without loss of generality, we can assume $a > b$ and $c > d$.
2. For the expression to be an integer, the denominator $2^c - 2^d$ must divide the numerator $2^a - 2^b$. We can factor both the numerat... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$2019$ circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours? | 1. **Assigning Numbers to Regions**:
For each region \(i\), assign a number \(a_i\) such that all points in region \(i\) are contained inside exactly \(a_i\) of the 2019 circles. This number \(a_i\) represents the count of circles that enclose the region \(i\).
2. **Parity Argument**:
We claim that no region \(... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alice and Bob are independently trying to figure out a secret password to Cathy’s bitcoin wallet. Both of them have already figured out that:
$\bullet$ it is a $4$-digit number whose first digit is $5$.
$\bullet$ it is a multiple of $9$;
$\bullet$ The larger number is more likely to be a password than a smaller number... | 1. **Identify the constraints:**
- The number is a 4-digit number starting with 5.
- The number is a multiple of 9.
- Alice knows the second and third digits.
- Bob knows the third and fourth digits.
- Both initially have no idea what the number is.
- The larger number is more likely to be the passwor... | 5940 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For how many integers $n$ between $ 1$ and $2021$ does the infinite nested expression $$\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}}$$ give a rational number? | 1. Let the infinite nested expression be denoted by \( x \). Therefore, we have:
\[
x = \sqrt{n + \sqrt{n + \sqrt{n + \sqrt{\ldots}}}}
\]
Squaring both sides, we get:
\[
x^2 = n + \sqrt{n + \sqrt{n + \sqrt{\ldots}}}
\]
Since the expression inside the square root is the same as \( x \), we can wr... | 44 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Each of the $10$ dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice crea... | 1. Let's define the variables as follows:
- \( v_T \): Number of dwarfs who like vanilla ice cream and tell the truth.
- \( v_F \): Number of dwarfs who like vanilla ice cream and lie.
- \( c_T \): Number of dwarfs who like chocolate ice cream and tell the truth.
- \( c_F \): Number of dwarfs who like choco... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer is called sparkly if it has exactly 9 digits, and for any n between 1 and 9 (inclusive), the nth digit is a positive multiple of n. How many positive integers are sparkly? | To determine how many positive integers are sparkly, we need to analyze each digit's constraints based on its position. A sparkly number has exactly 9 digits, and for any \( n \) between 1 and 9 (inclusive), the \( n \)-th digit must be a positive multiple of \( n \).
1. **First Digit (n=1):**
- The first digit can... | 216 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers less than $2019$ are divisible by either $18$ or $21$, but not both? | 1. **Calculate the number of integers less than 2019 that are divisible by 18:**
\[
\left\lfloor \frac{2018}{18} \right\rfloor = \left\lfloor 112.1111 \right\rfloor = 112
\]
So, there are 112 positive integers less than 2019 that are divisible by 18.
2. **Calculate the number of integers less than 2019 tha... | 176 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{x_1, x_2, x_3, ..., x_n\}$ be a set of $n$ distinct positive integers, such that the sum of any $3$ of them is a prime number. What is the maximum value of $n$? | 1. **Assume $n \geq 5$ and derive a contradiction:**
- Let $\{x_1, x_2, x_3, \ldots, x_n\}$ be a set of $n$ distinct positive integers such that the sum of any three of them is a prime number.
- Consider the residues of these integers modulo 3. There are three possible residues: 0, 1, and 2.
2. **Pigeonhole Prin... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$. | 1. **Define the sequence and initial conditions:**
The sequence \( x_1, x_2, x_3, \ldots \) is defined by \( x_1 = 2022 \) and \( x_{n+1} = 7x_n + 5 \) for all positive integers \( n \).
2. **Express the given expression in combinatorial form:**
The expression to analyze is:
\[
\frac{x_n(x_n - 1)(x_n - 2) ... | 404 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of the smallest and largest possible values for $x$ which satisfy the following equation.
$$9^{x+1} + 2187 = 3^{6x-x^2}.$$ | 1. Start with the given equation:
\[
9^{x+1} + 2187 = 3^{6x - x^2}
\]
2. Rewrite \(9^{x+1}\) and \(2187\) in terms of base 3:
\[
9^{x+1} = (3^2)^{x+1} = 3^{2(x+1)} = 3^{2x+2}
\]
\[
2187 = 3^7
\]
3. Substitute these into the original equation:
\[
3^{2x+2} + 3^7 = 3^{6x - x^2}
\]
4.... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
$AB$ is a chord of length $6$ in a circle of radius $5$ and centre $O$. A square is inscribed in the sector $OAB$ with two vertices on the circumference and two sides parallel to $ AB$. Find the area of the square. | 1. **Construct the initial square:**
- Consider the square $ABCD$ with side length $6$ such that $C$ and $D$ lie outside the circle.
- Since $OC = OD$ by symmetry, there exists a homothety about $O$ that maps $C$ and $D$ to $C'$ and $D'$ such that $C'$ and $D'$ lie on the circle.
2. **Calculate the distance $OC$... | 36 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime). | 1. **Consider the problem statement**: We need to find all primes \( p \) that can be written both as a sum and as a difference of two primes.
2. **Set up the equations**: Let \( p \) be a prime such that:
\[
p = x + y \quad \text{and} \quad p = z - w
\]
where \( x, y, z, w \) are primes.
3. **Consider t... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given $2014$ points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three connected points? | 1. **Interpret the problem in terms of graph theory:**
- We are given 2014 points in the plane, no three of which are collinear.
- We need to find the minimum number of line segments (edges) such that adding any additional line segment will always form a triangle (a $K_3$ subgraph).
2. **Graph Theory Translation... | 2013 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We are given a $4\times4$ square, consisting of $16$ squares with side length of $1$. Every $1\times1$ square inside the square has a non-negative integer entry such that the sum of any five squares that can be covered with the figures down below (the figures can be moved and rotated) equals $5$. What is the greatest n... | 1. **Understanding the Problem:**
We are given a $4 \times 4$ grid where each cell contains a non-negative integer. We need to ensure that the sum of any five cells that can be covered by the given figures equals 5. The figures can be moved and rotated.
2. **Analyzing the Figures:**
The figures are not explicitl... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
An equilateral triangle $T$ with side length $2022$ is divided into equilateral unit triangles with lines parallel to its sides to obtain a triangular grid. The grid is covered with figures shown on the image below, which consist of $4$ equilateral unit triangles and can be rotated by any angle $k \cdot 60^{\circ}$ for... | To solve this problem, we need to determine the smallest possible number of figures of type 1 that can be used to cover an equilateral triangle \( T \) with side length 2022, divided into unit equilateral triangles.
1. **Calculate the total number of unit triangles in \( T \)**:
The number of unit triangles in an e... | 1011 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $1000$ students in a school. Every student has exactly $4$ friends. A group of three students $ \left \{A,B,C \right \}$ is said to be a [i]friendly triplet[/i] if any two students in the group are friends. Determine the maximal possible number of friendly triplets.
[i]Proposed by Nikola Velov[/i] | 1. **Graph Interpretation**:
- Consider the problem in terms of graph theory. Let \( G \) be a graph where each vertex represents a student and each edge represents a friendship.
- Given that there are \( n = 1000 \) students and each student has exactly \( d = 4 \) friends, \( G \) is a 4-regular graph with 1000... | 2000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x,y$ and $z$ be positive real numbers such that $xy+z^2=8$. Determine the smallest possible value of the expression $$\frac{x+y}{z}+\frac{y+z}{x^2}+\frac{z+x}{y^2}.$$ | 1. Given the constraint \(xy + z^2 = 8\), we need to find the minimum value of the expression:
\[
\frac{x+y}{z} + \frac{y+z}{x^2} + \frac{z+x}{y^2}.
\]
2. To approach this problem, we can use the method of Lagrange multipliers or apply inequalities such as the AM-GM inequality. Here, we will use the AM-GM ine... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant $X$, let $t(X)$ be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true:
$1)$ For any two friends $X'$... | 1. **Graph Representation**:
- Consider a graph \( G \) where each vertex represents a contestant.
- There is an edge between two vertices \( A \) and \( B \) if and only if the contestants \( A \) and \( B \) are friends.
- Each vertex \( A \) is assigned a value \( t(A) \), which is the total score that cont... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest possible area of a quadrilateral with sidelengths $1, 4, 7$ and $8$ ? | 1. **Identify the problem and given data:**
We need to find the largest possible area of a quadrilateral with side lengths \(1, 4, 7,\) and \(8\).
2. **Use Bretschneider's formula:**
The area \(A\) of a general quadrilateral with sides \(a, b, c, d\) and angles \(\alpha\) and \(\gamma\) opposite each other is gi... | 18 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$.
(a) Show that $s(n)$ is an integer whenever $n$ is an integer.
(b) How many integers $n$ with $0 < n \le 2008$ are such that $s(n)$ is divisible by $4$?
| 1. **Show that \( s(n) \) is an integer whenever \( n \) is an integer.**
We start with the given function:
\[
s(n) = \frac{1}{6} n^3 - \frac{1}{2} n^2 + \frac{1}{3} n
\]
Factor out \( n \):
\[
s(n) = n \left( \frac{1}{6} n^2 - \frac{1}{2} n + \frac{1}{3} \right)
\]
Combine the terms inside ... | 1255 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Tadeo draws the rectangle with the largest perimeter that can be divided into $2015$ squares of sidelength $1$ $cm$ and the rectangle with the smallest perimeter that can be divided into $2015$ squares of sidelength $1$ $cm$. What is the difference between the perimeters of the rectangles Tadeo drew? | 1. **Understanding the problem**:
- We need to find the difference between the perimeters of the largest and smallest rectangles that can be divided into 2015 squares of side length 1 cm.
- The area of each rectangle is \(2015 \, \text{cm}^2\) since each square has an area of \(1 \, \text{cm}^2\).
2. **Finding t... | 3840 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Camila creates a pattern to write the following numbers:
$2, 4$
$5, 7, 9, 11$
$12, 14, 16, 18, 20, 22$
$23, 25, 27, 29, 31, 33, 35, 37$
$…$
Following the same pattern, what is the sum of the numbers in the tenth row? | 1. **Identify the pattern in the sequence:**
- The first row has 2 numbers: \(2, 4\)
- The second row has 4 numbers: \(5, 7, 9, 11\)
- The third row has 6 numbers: \(12, 14, 16, 18, 20, 22\)
- The fourth row has 8 numbers: \(23, 25, 27, 29, 31, 33, 35, 37\)
- We observe that the \(n\)-th row has \(2n\) n... | 3840 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider numbers of the form $1a1$, where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?
[i]Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$, $91719$.[/i] | 1. Consider numbers of the form \(1a1\), where \(a\) is a digit from 0 to 9. We need to find pairs of such numbers whose sum is a palindrome.
2. Let the two numbers be \(1a1\) and \(1b1\). Their sum is:
\[
1a1 + 1b1 = (100 + 10a + 1) + (100 + 10b + 1) = 200 + 10a + 10b + 2 = 202 + 10(a + b)
\]
3. For the sum t... | 55 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The rectangle in the figure has dimensions $16$ x $20$ and is divided into $10$ smaller equal rectangles. What is the perimeter of each of the $10$ smaller rectangles? | 1. **Identify the dimensions of the large rectangle:**
The large rectangle has dimensions \(16 \times 20\).
2. **Determine the area of the large rectangle:**
\[
\text{Area of the large rectangle} = 16 \times 20 = 320
\]
3. **Determine the area of each smaller rectangle:**
Since the large rectangle is d... | 24 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are $12$ dentists in a clinic near a school. The students of the $5$th year, who are $29$, attend the clinic. Each dentist serves at least $2$ students. Determine the greater number of students that can attend to a single dentist . | 1. We start by noting that there are 12 dentists and 29 students. Each dentist serves at least 2 students. Therefore, the minimum number of students served by all dentists is:
\[
12 \times 2 = 24 \text{ students}
\]
2. Since there are 29 students in total, the number of students left after each dentist serves ... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all natural numbers $n$ for which $n + 195$ and $n - 274$ are perfect cubes. | 1. Let \( n + 195 = x^3 \) and \( n - 274 = y^3 \). This implies:
\[
n = x^3 - 195 \quad \text{and} \quad n = y^3 + 274
\]
Equating the two expressions for \( n \), we get:
\[
x^3 - 195 = y^3 + 274
\]
Simplifying, we obtain:
\[
x^3 - y^3 = 469
\]
2. We can factorize \( x^3 - y^3 \) usi... | 2002 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In the rectangular parallelepiped in the figure, the lengths of the segments $EH$, $HG$, and $EG$ are consecutive integers. The height of the parallelepiped is $12$. Find the volume of the parallelepiped.
[img]https://cdn.artofproblemsolving.com/attachments/6/4/f74e7fed38c815bff5539613f76b0c4ca9171b.png[/img] | 1. We start by noting that $\triangle EHG$ is a right triangle. The lengths of the segments $EH$, $HG$, and $EG$ are consecutive integers. Since $EG$ is the hypotenuse, we can use the Pythagorean theorem to find the relationship between these sides.
2. Let the lengths of $EH$, $HG$, and $EG$ be $a$, $a+1$, and $a+2... | 144 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle $ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/9/2/2013de890e438f5bf88af446692b4... | 1. Let the cevians intersect at point \( F \). The cevians divide triangle \( ABC \) into six smaller triangles. We are given the areas of four of these triangles: \( [AFY] = 84 \), \( [CFX] = 70 \), \( [BFX] = 35 \), and \( [BFY] = 65 \).
2. Let the areas of the remaining two triangles be \( x \) and \( y \). Specifi... | 380 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
With the digits $1, 2, 3,. . . . . . , 9$ three-digit numbers are written such that the sum of the three digits is $17$. How many numbers can be written? | To solve the problem, we need to count the number of three-digit numbers formed by the digits \(1, 2, 3, \ldots, 9\) such that the sum of the digits is 17. We can use generating functions to solve this problem.
1. **Formulate the Generating Function:**
Each digit \(d_i\) (where \(i = 1, 2, 3\)) can take any value f... | 57 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three [i]aligned[/i] numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$? | 1. **Define the problem and variables:**
We are given the set $\{1, 2, 3, 4, 5, 6, 7\}$ and need to place each number in a circle such that the sum of any three aligned numbers is the same. Let this common sum be denoted by $s$.
2. **Set up the equations:**
Consider the three vertical columns:
\[
(A + D + ... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small.
[img]https://cdn.artofproblemsolving.com/attachments/a/1/aedfbfb2cb17bf816aa7daeb0d35f46a79b6e9.jpg[/img]
Nair build triangular figures by following these rules:
$\bullet$ Figure $1$ is made up of $4$ small pieces,... | 1. **Identify the pattern for small triangles ($S_n$):**
- For Figure 1: $S_1 = 4$
- For Figure 2: $S_2 = 8$
- For Figure 3: $S_3 = 12$
- Observing the pattern, we see that the number of small triangles increases by 4 for each subsequent figure. Thus, we can generalize this as:
\[
S_n = 4n
\]... | 460 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $13$ positive integers greater than $\sqrt{15}$ and less than $\sqrt[3]{B}$. What is the smallest integer value of $B$? | 1. We need to find the smallest integer value of \( B \) such that there are 13 positive integers greater than \( \sqrt{15} \) and less than \( \sqrt[3]{B} \).
2. First, calculate \( \sqrt{15} \):
\[
\sqrt{15} \approx 3.872
\]
Therefore, the smallest integer greater than \( \sqrt{15} \) is 4.
3. Let \( a ... | 4097 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ such that $n$ is divisible by all the positive integers less than $\sqrt[3]{n}$. | 1. Let \( x \) be the largest integer less than \( \sqrt[3]{n} \). This implies \( x \leq \sqrt[3]{n} < x+1 \), and thus \( n \leq (x+1)^3 \).
2. Since \( n \) must be divisible by all positive integers less than \( \sqrt[3]{n} \), \( n \) must be divisible by all integers from 1 to \( x \). Therefore, \( n \) must be... | 420 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $a$ and $b$ are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of $p$? | 1. Given that \( p = \frac{b}{4} \sqrt{\frac{2a - b}{2a + b}} \) is a prime number, we need to find the maximum possible value of \( p \).
2. First, we note that if \( b \) is odd, then \( p \) is not an integer. Therefore, \( b \) must be even. Let \( b = 2c \) for some positive integer \( c \).
3. Substituting \( b =... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$ | 1. Let \( N = 100a + 10b + c \), where \( a, b, c \in \{0, 1, \ldots, 9\} \). Since \( N \) is a three-digit number, \( a \neq 0 \).
2. Given that \( N \) is divisible by 11, we use the divisibility rule for 11: the alternating sum of the digits must be divisible by 11. Therefore, \( a - b + c \equiv 0 \pmod{11} \). T... | 550 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 1. We start with the given equations:
\[
15a + 16b = x^2 \quad \text{and} \quad 16a - 15b = y^2
\]
where \(x\) and \(y\) are positive integers.
2. We add and subtract these equations to eliminate \(b\) and \(a\) respectively:
\[
(15a + 16b) + (16a - 15b) = x^2 + y^2 \implies 31a + b = x^2 + y^2
\]... | 231361 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine all positive integers $n$ such that $ xy+1 \equiv 0 \; \pmod{n} $ implies that $ x+y \equiv 0 \; \pmod{n}$. | To determine all positive integers \( n \) such that \( xy + 1 \equiv 0 \pmod{n} \) implies \( x + y \equiv 0 \pmod{n} \), we proceed as follows:
1. **Assume \( xy + 1 \equiv 0 \pmod{n} \)**:
\[
xy \equiv -1 \pmod{n}
\]
2. **Consider \( x \) and \( y \) such that \( x \) is coprime to \( n \)**:
Since \( ... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$ be an odd prime of the form $p=4n+1$. [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$. [*] Calculate the value $n^{n}$ $\pmod{p}$. [/list] | 1. **Show that \( n \) is a quadratic residue \(\pmod{p}\):**
Given \( p = 4n + 1 \), we need to show that \( n \) is a quadratic residue modulo \( p \).
Since \( p \equiv 1 \pmod{4} \), we know that \( -1 \) is a quadratic residue modulo \( p \). This means there exists an integer \( i \) such that \( i^2 \equ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest prime which is not the difference (in some order) of a power of $2$ and a power of $3$. | To find the smallest prime which is not the difference (in some order) of a power of $2$ and a power of $3$, we need to check each prime number and see if it can be expressed as $2^x - 3^y$ or $3^y - 2^x$ for non-negative integers $x$ and $y$.
1. **Check small primes:**
- $2 = 3^1 - 2^0$
- $3 = 2^2 - 3^0$
- $... | 41 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
One of Euler's conjectures was disproved in the $1980$s by three American Mathematicians when they showed that there is a positive integer $n$ such that \[n^{5}= 133^{5}+110^{5}+84^{5}+27^{5}.\] Find the value of $n$. | 1. **Divisibility by 2:**
Since 110 and 84 are even, and 27 and 133 are odd, the sum of their fifth powers will be even. Therefore, \( n^5 \) is even, implying \( n \) is even. Thus, \( 2 \mid n \).
2. **Divisibility by 3:**
Both 84 and 27 are divisible by 3. We need to check if \( 133^5 + 110^5 \) is divisible ... | 144 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A function $f$ is defined on the positive integers by \[\left\{\begin{array}{rcl}f(1) &=& 1, \\ f(3) &=& 3, \\ f(2n) &=& f(n), \\ f(4n+1) &=& 2f(2n+1)-f(n), \\ f(4n+3) &=& 3f(2n+1)-2f(n), \end{array}\right.\] for all positive integers $n$. Determine the number of positive integers $n$, less than or equal to 1988, for ... | 1. **Define the function and initial values:**
The function \( f \) is defined on the positive integers by:
\[
\begin{cases}
f(1) = 1, \\
f(3) = 3, \\
f(2n) = f(n), \\
f(4n+1) = 2f(2n+1) - f(n), \\
f(4n+3) = 3f(2n+1) - 2f(n),
\end{cases}
\]
for all positive integers \( n \).
2. **Prove... | 92 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The function $f: \mathbb{N}\to\mathbb{N}_{0}$ satisfies for all $m,n\in\mathbb{N}$: \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine $f(1982)$. | 1. **Given Conditions and Initial Analysis:**
- The function \( f: \mathbb{N} \to \mathbb{N}_0 \) satisfies:
\[
f(m+n) - f(m) - f(n) = 0 \text{ or } 1
\]
for all \( m, n \in \mathbb{N} \).
- Additional conditions:
\[
f(2) = 0, \quad f(3) > 0, \quad \text{and} \quad f(9999) = 3333.
... | 660 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_{1}={11}^{11}$, $a_{2}={12}^{12}$, $a_{3}={13}^{13}$, and \[a_{n}= \vert a_{n-1}-a_{n-2}\vert+\vert a_{n-2}-a_{n-3}\vert, n \ge 4.\] Determine $a_{{14}^{14}}$. | 1. We start by defining the sequence \(a_n\) as given:
\[
a_1 = 11^{11}, \quad a_2 = 12^{12}, \quad a_3 = 13^{13}
\]
and for \(n \geq 4\),
\[
a_n = |a_{n-1} - a_{n-2}| + |a_{n-2} - a_{n-3}|
\]
2. Define the difference \(\Delta a_n = a_{n+1} - a_n\). Then, we can express the recurrence relation in ... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence? | To determine the maximal number of successive odd terms in a sequence where each term is obtained by adding its largest digit to the previous term, we need to analyze the behavior of the sequence based on the last digit of each term.
1. **Initial Considerations**:
- If a term ends in 9, the next term will be even b... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In the sequence $00$, $01$, $02$, $03$, $\cdots$, $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$, $39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could rem... | 1. **Define Digit Parity**: The digit parity of a number is defined as the parity (odd or even) of the sum of its digits. For example, the digit parity of 29 is the parity of \(2 + 9 = 11\), which is odd.
2. **Sequence Rearrangement**: Let \(a_1, a_2, \ldots, a_{100}\) be the rearranged sequence. We need to show that ... | 50 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $S=\{1,2,3,\ldots,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime. | To solve the problem, we need to find the smallest integer \( n \) such that any \( n \)-element subset of \( S = \{1, 2, 3, \ldots, 280\} \) contains five numbers that are pairwise relatively prime.
1. **Identify the subset \( A \) of multiples of 2, 3, 5, or 7:**
- Calculate the number of multiples of each prime ... | 217 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest integer $n \ge 4$ for which one can choose four different numbers $a, b, c, $ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$ . | To determine the smallest integer \( n \ge 4 \) for which one can choose four different numbers \( a, b, c, \) and \( d \) from any \( n \) distinct integers such that \( a + b - c - d \) is divisible by \( 20 \), we can use the Pigeonhole Principle (PHP).
1. **Understanding the Problem:**
We need to find the small... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that [list][*] each member in $S$ is a positive integer not exceeding $2002$, [*] if $a,b\in S$ (not necessarily different), then $ab\not\in S$. [/list] | 1. **Define the problem constraints and initial assumptions:**
- We need to find the largest positive integer \( n \) such that there exists a set \( S \) with exactly \( n \) numbers.
- Each member in \( S \) is a positive integer not exceeding \( 2002 \).
- If \( a, b \in S \) (not necessarily different), th... | 1958 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What's the largest number of elements that a set of positive integers between $1$ and $100$ inclusive can have if it has the property that none of them is divisible by another? | 1. **Representation of Numbers**:
Each positive integer \( n \) can be uniquely written as \( n = 2^k \cdot q \) where \( k \) is a non-negative integer and \( q \) is an odd positive integer. This is essentially the prime factorization of \( n \) where \( q \) is the product of all odd prime factors of \( n \).
2.... | 50 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a subset of $\{1, 2, 3, \cdots, 1989 \}$ in which no two members differ by exactly $4$ or by exactly $7$. What is the largest number of elements $S$ can have? | 1. **Define the problem and constraints:**
We need to find the largest subset \( S \) of \(\{1, 2, 3, \ldots, 1989\}\) such that no two elements in \( S \) differ by exactly \( 4 \) or \( 7 \).
2. **Identify the structure of the problem:**
Consider the set \(\{1, 2, 3, \ldots, 1989\}\). We need to ensure that fo... | 905 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A finite sequence of integers $a_{0}, a_{1}, \cdots, a_{n}$ is called quadratic if for each $i \in \{1,2,\cdots,n \}$ we have the equality $\vert a_{i}-a_{i-1} \vert = i^2$. [list=a] [*] Prove that for any two integers $b$ and $c$, there exists a natural number $n$ and a quadratic sequence with $a_{0}=b$ and $a_{n}=c$.... | ### Part (a)
1. **Restate the problem**: We need to prove that for any two integers \( b \) and \( c \), there exists a natural number \( n \) and a quadratic sequence \( a_0, a_1, \ldots, a_n \) such that \( a_0 = b \) and \( a_n = c \).
2. **Define the sequence**: A sequence is quadratic if for each \( i \in \{1, 2,... | 19 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that \[\prod_{n=1}^{1996}(1+nx^{3^{n}}) = 1+a_{1}x^{k_{1}}+a_{2}x^{k_{2}}+\cdots+a_{m}x^{k_{m}}\] where $a_{1}$, $a_{2}$,..., $a_{m}$ are nonzero and $k_{1}< k_{2}< \cdots < k_{m}$. Find $a_{1996}$. | 1. We start with the given product:
\[
\prod_{n=1}^{1996}(1+nx^{3^{n}}) = 1 + a_{1}x^{k_{1}} + a_{2}x^{k_{2}} + \cdots + a_{m}x^{k_{m}}
\]
where \(a_{1}, a_{2}, \ldots, a_{m}\) are nonzero coefficients and \(k_{1} < k_{2} < \cdots < k_{m}\).
2. Each \(k_i\) is a sum of distinct powers of 3, i.e., \(k_i = 3... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the maximum value of $m^{2}+n^{2}$, where $m$ and $n$ are integers satisfying $m,n\in \{1,2,...,1981\}$ and $(n^{2}-mn-m^{2})^{2}=1.$ | To determine the maximum value of \( m^2 + n^2 \), where \( m \) and \( n \) are integers satisfying \( m, n \in \{1, 2, \ldots, 1981\} \) and \((n^2 - mn - m^2)^2 = 1\), we need to solve the equation \((n^2 - mn - m^2)^2 = 1\).
This equation can be rewritten as:
\[ n^2 - mn - m^2 = \pm 1 \]
We will consider both cas... | 3524578 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Alice and Bob play the following number-guessing game. Alice writes down a list of positive integers $x_{1}$, $\cdots$, $x_{n}$, but does not reveal them to Bob, who will try to determine the numbers by asking Alice questions. Bob chooses a list of positive integers $a_{1}$, $\cdots$, $a_{n}$ and asks Alice to tell him... | **
One round does not suffice because if Bob only asks for one sum, there could be multiple sets of \( x_i \) that produce the same sum. For example, if Alice's numbers are such that the result can be represented in more than one way as an integral linear combination of the given \( a_k \), Bob would not be able to ... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The set $M= \{1;2;3;\ldots ; 29;30\}$ is divided in $k$
subsets such that if $a+b=n^2, (a,b \in M, a\neq b, n$ is an
integer number $)$, then $a$ and $b$ belong different subsets.
Determine the minimum value of $k$. | 1. **Identify the pairs \((a, b)\) such that \(a + b = n^2\):**
We need to find all pairs \((a, b)\) in the set \(M = \{1, 2, 3, \ldots, 30\}\) such that \(a + b = n^2\) for some integer \(n\). The possible values of \(n^2\) within the range of \(M\) are \(4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, ... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.